Q  A    J 

^^   '^       ll  UC-NRLF 

S     I        $B    532    fl31 


EXCHANOB 

JlIN  14  1917 


DIFFERENTIAL  INVARIANTS 
UNDER  THE  INVERSION  GROUP 


BY 


GEORGE  WALKER  MULLINS 


Submitted  in  Partial  Fulfilment  of  the  Requirements 

FOR  the  Degree  of  Doctor  of  Philosophy  in 

the  Faculty  op  Pure  Science, 

Columbia  University 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY 

LANCASTER.  PA. 

1917 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

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DIFFERENTIAL  INVARIANTS 
UNDER  THE  INVERSION  GROUP 


BY 


GEORGE  WALKER  MULLINS 


Submitted  in  Partial  Fulfilment  of  the  Requirements 

FOR  the  Degree  of  Doctor  of  Philosophy  in 

THE  Faculty  of  Pure  Science, 

Columbia  University 


PRESS  OF 

THE  NEW  ERA  PRfNTINQ  COMPANY 

LANCASTER.  PA. 

1917 


-MS 


7   ■!l?<'  •     " 


DIFFERENTIAL   INVARIANTS   UNDER   THE    INVER- 
SION GROUP. 

Introduction. 

The  object  of  this  paper  is  to  study  the  differential  invariants 
that  arise  under  the  continuous  transformation  group  of  six 
parameters,  known  as  the  inversion  group  of  the  plane,  and 
given  by  the  equations  of  transformation 

(Q!iYi+Q!272)(a:^+y^)  +  («i+/3m+j32T2)a;-(ata+ffi72-/327i)y+/9i 


X  = 
F  = 


(7i''+72^)(x2+2/2)+2TiX-272y+l 
(a27i  -aiYz)  (x^+y^)  +  (a2+/327i  -/3i72)x  +  (ai— /3i7i  -/3272)y+i8s 


(7i^+72')(x2+2/2)+27ia;-272j/+l 

Inversion  geometry  for  the  most  part  has  been  studied  syn- 
thetically, in  connection  with  the  theory  of  functions  of  a  complex 
variable  satisfying  the  general  linear  fractional  transformation 
group 

72  +  0 

Thus  it  is  known  that  the  families  of  minimal  lines,  circles,  and 
logarithmic  equiangular  double  spirals  remain  invariant  under 

Ge.t 

The  justification  for  the  study  of  the  group  from  the  stand- 
point of  differential  geometry,  aside  from  the  classification  or 
codification  of  the  invariants  under  the  group,  is  the  bringing  to 
light  of  some  new  properties  of  the  logarithmic  double  spirals. 

In  part  one,  the  Lie  theory  is  employed  in  finding  the  differ- 
ential invariant  (I5)  of  lowest  order,  while  in  part  two  the  inter- 
pretation of  this  invariant  is  given  by  making  use  of  a  certain 
family  of  bicircular  quartic  curves.  Furthermore  the  differential 
equation  of  the  logarithmic  equiangular  double  spirals  is  found 

*See  Holzmiiller:  Theorie  der  Isogonal  Verwandenschaften;  also  Cole, 
"  Linear  Functions  of  a  Complex  Variable,"  Annals  of  Mathematics,  Vol.  V: 
No.  4. 

t  Throughout  this  paper  we  shall  refer  to  the  inversion  group  as  Gt. 

3 


360599 


4      . '-. ": :/:  V'.  Difmrential  In\^ariants  under 

and  expressed  in  terms  of  1 5,  and  from  this  differential  equation, 
certain  properties  of  the  family  are  deduced. 

Part  three  is  concerned  with  differential  invariants  of  order  six. 
Here  again  other  properties  of  the  double  spiral  are  found. 
Finally,  part  four  deals  with  differential  invariants,  intrinsically 
expressed,  and  with  the  classification  of  invariants,  of  order  higher 
than  the  sixth. 

I.    The  Absolute  Differential  Invariant  of  Lowest  Order. 

1.  The  six  independent  infinitesimal  transformations  gener- 
ating the  inversion  group,  Ge,  may  be  taken  as, 

A  solution  of  the  complete  system  [2]  of  differential  equations 
formed  by  equating  to  zero,  each  of  the  above  five-fold  extended 
independent  infinitessimal  transformations  of  G^  is  a  differential 
invariant  of  lowest  order  under  Ge*  Extending  the  infinites- 
imal transformations  Uif  [i  =  1,  2,  3,  4,  5,  6],  fivefold  and 
equating  each  to  zero  we  have  for 

E 

Bx      "'    dy        ■ 
*  See:  Lie-Scheffers,  Continuierliche  Gruppen. 


THE  Inversion  Group.  5 

"" dx^y dy^^ dy'     y   dy"     ^^     dy'"     "^^    di/-     ^^  dy^      "' 
^2^'  "  ^'^  ^  "  ^"^2/  fy  -  %(!  +  2/")  ^  +  [2^2/"  -  6y2/'2/" 
-2y'{\+y'')^,  +  {W''-^yy'''-SyyY'-'^'^y'Y]^, 

+  [80:2/-  +  lOi/^^  -  I2yy'y-  -  SOyy'Y^  -  SOy'Y^ 

-  myyy"  -  20yy""  -  dOy"^^  =  0, 

2^2/^  +  if  -  ^')  ^  -  2x(l  +  y'')  f-,  -  [2yy"  +  Qxy'y" 


n]  ^  -  [42/2/-+  6a:2,"^+8a^2/'2/-'+122/'^"]  ^ 


/..///21     ^/ 


-  [62/2/-  +  lOxyY-  +  182/"2  +  20a:2/"2/'"  +  24yy"']  ^^ 

-  [182/2/^  +  I2xy'y^  +  40z/'2/'^  +  mxy"y'^  +  SOy"y"' 


jn2,   ^f 


-^20xy'"\^^  =  0. 

In  order  to  obtain  a  solution  of  the  complete  system  of  differential 
equations  2,  in  the  simplest  manner,  it  will  be  necessary  to  find 
first  the  differential  invariants  under  the  Hauptgruppe.* 

2.  Let  us  make  use  of  Lies'  differential  parameter  theory  and 
assume  that  ^  is  a  differential  invariant  under  the  Hauptgruppe. 
By  extending  each  of  the  infinitesimal  transformations  Ui{f) 
[i  =  1,  2,  3,  4]  three-fold  and  then  with  respect  to  <p  and  equating 
each  to  zero,  we  have  a  complete  system  of  linear  partial  differen- 

*  lOein  has  called  the  group  generated  by  df/dx,  df/dy,  —  y{df/dx)  +x{df/dy), 
xidf/dx)  +  yidf/dy),  the  Hauptgruppe. 


6  Differential  Invariants  under 

tial  equations 


1=0. 


+  [4yy"  +  32,"1  ^, . . .  +  2/ V  ^  =  0, 

having  two  independent  solutions.    Since 

dx      dy        * 
the  system  S  reduces  to 

5': 


y   dy"      ^y     dy'"      'f  d<p'~^' 


and  the  two  independent  solutions  of  S'  are 

3yy^'  -  y"'il  +  y") 

ym  -  c, 

and 

(1  +  y'^<p' 

77 =  ^2. 

y 

These  two  invariants*  expressed  in  terms  of  the  radius  of  cur- 
vature, p,  the  arc  length  s,  and  the  derivative  of  p  with  respect 
to  8  are 

dp  d<p 

d^  =  '''    Pdl  =  '" 

*  It  is  interesting  to  note  that  the  Hauptgruppe  may  be  expressed  as  a  one 
parameter  intrinsic  transformation  group  whose  infinitesimal  transformation 
is  p{df/dp)  +  8{dfjds)  and  the  invariants  above  are  readily  obtained  from  this. 


THE  Inversion  Group.  7 

Since  the  differential  invariant  of  lowest  order  is  known,  i.  e., 
dp/ds,  it  follows  at  once,  by  making  use  of  the  differential  para- 
meter p{d<pfds),  that  those  of  higher  successive  orders  are 

dp  ^         2^  n-3^li 

ds'  ^ds"  ^  ds^'   '"'  ^      ds^-^' 

From  the  preceding  discussion  we  may  conclude: 

Theorem  I.     The  most  general  differential  invariant  of  order  n 
under  the  Hauptgruppe  is  of  the  form 


f 


[dp      ^^       ...      ,n-3^Pl 

Ids'^ds^'        '^      ds^-^]' 


Moreover  since  the  Hauptgruppe  is  a  subgroup  of  Ge,  we  have: 
Corollary  I.    All  differential  invariants  under  Gt  must  he  of 
the  form 


3.  Let  us  now  return  to  the  system  S  of  section  one.    Since 

dx      dy        * 
2  becomes 

(1  +  y'')  |i  +  Sy'y"  ^  +  Wy'"  +  Sy"']  j^, 

+  [52/Vv+102/'y'1^ 

+  [62/y  +  ir>y"t-  +  \^y"J'\  ^  =  0, 

y   dy"      ^    dy'"        y    dy'^        ^  dy-      ^' 
-  22/(1  +  y")  ^  +  \2xy"  -  ^yy'y"  -  2y'{\  +  y")]  ^ 
+  [Axy'"  -  Gyy'"  -  SyyY"  -  l2y'Y]   ^^ 


dy'" 
2J':      +  [Qxt-  -  lOyy'r-  -  20y'Y'  +  4y"']  ^ 


Differential  Invariants  under 


+  [Sxy^  +  lOy'^  -  I2yy'y^  -  dOyy'Y''  —  SOy'Y^ 

-  UOy'y'Y'  -  ^Oyy""  -  30i/"^]^  =  0, 

-  2a:(l  +  y'')  ^  -  [2yy"  +  Qxy'y-  +  2(1  +  y")  ^ 
-  [W  +  ^xy'"  +  ^xy'y"'  +  l^y'y"]  ^, 


-  [18yr  +  l2xyY  +  40y'^^^  +  SOa;?/"?/-  +  SOy'Y 


and  S'  is  in  turn  reducible  to.* 


+  20xy'-']^^=0, 


-2/ 


.  ^/ 


S'^ 


[4y'"(l  +  2/'^  -  122/'2/"»] 


-  2y'"^ 


dy' 

dy'^ 


df  df 


+  [i02/-(i  +  y")  -  Wy"y"'  -  W"]^.  =  0, 

*  It  is  known  that  the  complete  system  of  linear  partial  differential  equa- 
tions, Ai(f)  =  0,  Aiif)  =0,  •  •  •,  Ar(f)  =  0,  has  the  same  solutions  as  the 
system 

Bkf  =  ftkiAJ  +  pLkzAsf  +  •  •  •  fXkrArf    [fc  =  1,  2,  3  •  •  •  r], 

where  the  fi'a  are  functions  of  the  variables  provided 


+  0. 


Serret-Scheffers:  Lehrbuch  der  Diff.  u.  Int.  Rech.,  Vol.  III.  If  aiss  =  m 
=  ms  =  ftw  =  1,  /XM  =  M«3  =  2z,  fiu  =  3y',  mss  =  M3«  =  ws  =  /^  =  M«  =  p« 
=  0,  MM  =  —  2y',  H63  =  2y,  uti  =  —  2y,  S'  reduces  to  S". 


Mil 

M12       • 

•       Mir 

M21 

M22        • 

•       M2r 

Mrl 

Mr2        • 

•        Mrr 

THE  Inversion  Group.  9 

-  2(1  +  y")  ^,  -  Uy'y"  ^,+  [AyV  -  2py'y"' 
-  \2yy"  -  W']^y-  [302/'y-  -  SOy'Y' 

-402/yy"-302/y'^]^=o. 

The  following  change  in  variables  in  S"  is  suggested  by  Corollary 

I,  section  two. 

Let 

zi  =  y',    22  =  y", 

dp     Sy'y'"  -  y"'{l  +  y") 
^'~ds~  y'" 

dp      (Sy"'  -  2y'y"')  d  +  y")      (v'Y''  -  ^y'"')  (1  + 1/")' 


^'-Pds-  y'"  y 

(Pp    Wy"'y'''  -  Qy""  -  y'Y)a  +  y")' 


//4 


ds^  y'" 

{y"Y'  +  ^y'y""  -  ^yy'r'')a  +  y")' 


+  y^n 


{2y'y"'  -  ^y'y"^{^  +  y") 

-t-  ym 

In  consequence  of  this  change  2"  becomes 

2z3^^+(5z4  + 223^)^^=0, 

2z3  ^^  +  (3z4  +  223^)  ^^  +  (4z5  +  6Z3Z4  -  223)  —^  =  0, 
and  reducing  2'"  to  its  equivalent  involution  system,  we  have 

423'^  +  (8Z3Z5  -   1524'  -  423^24  -  423*  -  523^)^  =   0. 

023  dZr, 

The  solution  of  S»^  is 

42325  —  524^  —  423^24  +  423*  —  423^ 


Z3 


C. 


10  Differential  Invariants  under 

If  we  pass  now  to  the  original  variables,  we  have  for  the  solution* 
of  the  system  S 

As'"''' 
where 

A3  =  ^y'y"  -  y"'{l  +  y") 
and 

A5  =  Wr  -  52/-=')(l  +  y"Y 

+  iWy'VY^  -  Wy"Y-\-2>0y"Y--4Qy'y""){l+yy 

+^Qy"Y'Ky"-  2)  {l+yy+  ISOyYY'a  -  y")  (1+2/'') 

-452/"«(l  +  Qy"-Sy% 

The  following  results  may  now  be  stated: 

Theorem  II.  The  first  absolute  differential  invariant  (the  one 
of  lowest  order)  under  Ge  is  of  order  five  and,  symbolically  expressed 
is  As/As^  =  h- 

For  symmetry  of  notation  let  us  write  the  differential  equation 
of  the  minimal  lines  1  -\-  y'^  =  0  as  Ai  =  0.  The  differential 
equation  of  the  circles  of  the  plane  is  A3  =  0.  We  may  think, 
of  Ai  and  A3  as  the  first  and  second  relative  differential  invariants 
under  Gq.    For  I5  =  0,  A5  must  be  zero.     Hence, 

Corollary  I.  The  third  relative  differential  invariant  under  Gt 
w  A5  =  0. 

In  part  two  the  integral  curves  of  A5  =  0  will  be  found. 

II.    The  Interpretation  of  h  and  the  Logarithmic  Equi- 
angular Double  Spirals. 

1.  For  an  interpretation  of  1 5  we  turn  to  the  family  of  bicircular 
quartic  curves,  given  by  the  equation 

Q:  {x'-^fy-{-{ax  +  by){x'+y')  +  cx'  +  2djcy  +  ef 

+  2fx  +  2gy+h  =  0. 

*  Of  course  any  function  of  As/As*  is  a  solution  of  S.  We  shall  however 
speak  of  the  simplest  form  of  a  solution,  as  '  the  '  solution.  Likewise  we  shall 
call  the  simplest  form  of  an  invariant  of  order  n  '  the  fundamental '  invariant 
of  order  n. 


THE  Inversion  Group.  11 

It  is  known  that  the  family  Q  remains  invariant  under  G^*  Let 
us  suppose  that  under  Ge  the  curve 

c:  y  =  cix  +  c^x^  +  Czx^  +  •  •  • 

is  transformed  into  the  curve 

C:  Y=  CiX  +  C2Z2  +  CzX'  H , 

where  both  c  and  C  are  assumed  to  be  analytic  in  the  neighborhood 
of  the  origin.  The  relation  between  the  coefficients  of  c  and  the 
successive  derivatives  at  the  point  in  question  is 

1 
1  .2  .3  ..-n^""^" 


and  similarly  for  C 

1 


1  .  2  .  3  •••  n 


r«  =  C„. 


lf,mQ,f=g=h  =  0,Q  has  a  double  point  at  the  origin  and 
its  equation  becomes 

Q':   (a;2  +  y^y  +  (ax  +  by){x''  +  y^)  +  ca^  +  2dxy  +  ey^  =  0. 

Since  there  are  five  independent  constants  in  Q',  one  member  of 
the  family  may  have  contact  of  order  five  with  c  at  the  origin 
P,  and  this  curve  will  serve  as  the  geometric  image  for  the  fifth 
derivative.  Expressing  the  fact  that  a  member  of  the  family 
Q'  may  osculate  c  at  P  we  have  Q'  subject  to  the  five  conditions 

1.  c+2y'd  +  y"e^0, 

2.  (1  +  y")a  +  2/'(l  +  y")b  +  y"d  +  yY'e  =  0, 

3.  12(1  +  yy+  I2y'y"a  +  W  +  I82/V  -  ]h 

-\-4y'''d+[Sy'"+4yy"]e  =  0, 

4.  2iy'y"a  +  y")  +  Wy"'  +  ^y'> 

+  W"  +  Qy'Y'  +  ^y'y"']h  +  y'^d  +  [2y'Y'  +  y'y'^]e  =  0, 

5.  240i/'2/'"(l  +  2/'^)  +  l^Qy"\l  +  Bi/'^)  +  {2,0yY- +  my"y"']a 

+  [152/-  +  45i/'V^"  +  l^Oyyy  +  452/"']6 

+  Qy-d  +  [15^''2/>v  +  IO2/""  +  Qyy]e  =  0. 
If  6  is  the  angle  between  the  two  branches  of  Q'  that  pass  through 

♦Casey:  On  Bicircular  Quartics,  Royal  Irish  Academy,  Transactions,  Vol. 
24. 


12  Differential  Invariants  under 

P  we  may  write 

(6)  Cot  0         "^  "^  ^ 


Vd2  -  ce  * 

Since  angles  are  preserved  under  G^  it  is  evident  that  ^  is  a 
function  of  As/As^.  On  solving  equations  1,  2,  3,  4,  5  for  c,  d 
and  e  and  substituting  in  (6)  we  have 

®  Cot0=-2^. 

From  Theorem  II  and  equation  (6')  we  have 
Theorem  III.    7/,  under  the  inversion  group  G^,  the  curve 

c:  y  =  cix  +  023^  +  c^pc^  +  •  •  • 

he  transformed  into  the  curve 

C:  Y=  CiX  +  C2Z2  +  CzX^  •  •  • 

{where  c  and  C  are  analytic  in  the  neighborhood  of  the  origin  P) 
the  angle  6  between  the  branches  of  the  osculating  bicircular  quartic 
having  a  double  point  at  P,  remains  invariant  and 

2.  It  is  known  that  the  family  of  the  00^  logarithmic  double 
spirals  of  the  plane  distribute  themselves  under  the  inversion 
group  into  00  ^  families  of  co^  curves  each,  each  one  of  the  co^ 
families  being  called  a  family  of  logarithmic  equiangular  double 
spirals.  In  such  a  family,  the  angle  between  the  polar  axis  and 
the  inflectional  tangent  is  the  same  for  each  curve  of  the  family. 
Intuitively  we  may  expect  Ag/Aa^  =  h  to  be  the  equation  of  the 
00*  logarithmic  double  spirals,  where  h  is  a  function  of  the  angle 
just  mentioned.  The  equation  of  the  logarithmic  double  spirals 
in  so-called  bipolar  coordinates  is  (7)  ri/r2  =  Vl-e^*.  If  the 
coordinates  of  the  poles  are  (a,  b)  and  (c,  d)  equation  (7)  expressed 
in  rectangular  cartesian  coordinates  becomes 

^7,^      {x-ay-^(y-by 


(x  -  cY  -\-  {y-  dy 
=  ke 


2A t>.n-i  {d-b)x+(a-c)y  +  bc-  ad 


{x'-{-f)-ia+c)x-ib+d)y  +  ac  +  bd' 


THE  Inversion  Group.  13 

If  we  put 

—  71  ,72 

a  =      o  . » .  0  = 


7i'  +  72''  7i'+722' 

^    —    (oClPl  4-  <X2^2)  ,  ^    —    («li32  —   CX2^l) 

ais  +  «2^  '  «1^  +  «2^ 


(7')  becomes 


,  _  71^  +  72'     2;,tan-i«27i  -  «172 
«i  +  «2''  aiji  +  a272 


^'^"^*   _  2A  tan-1  («2  7i  "  «i  72)  (a:*^  +  2/^)  +  («2  +  182  71  -  |8i  72)3; 


(«i2  +  a2^)  (a:2  +  2/2)  +  2  (ai  i8i  +  aa  (32) 

4-2(«i)82-«2/3i)  +  /3i'+/32« 

(yi'  +  y2')  {x'  +  2/2)  +  2^1  a;  -  22/1 2/  +  1 

■f  2/')  +  («2  +  i82  7i 

+  (q;i  —  )3i  7i  —  ^2  72)  y  +  /32 

(ai  7i  +  «2  72)  («^  +  y^)  +  («i  +  /3i  7i  +  i32  72)2:  ' 

—  (a2  +  )3i  72  —  02  7i)  2/  +  /3i 

which  is  a  form  more  suitable  for  the  purpose  of  using  the  curve 
for  contact. 

Since,  in  the  equation  of  a  logarithmic  equiangular  double 
spiral  (X  being  regarded  as  fixed),  there  are  five  independent  con- 
stants, the  curve  may  be  used  for  contact  of  the  fourth  order. 
Let  us  seek  the  condition  under  which  the  curve  may  hyperos- 
culate  the  curve  c  (section  1)  at  P.  Taking  the  successive  de- 
rivatives, as  far  as  the  fifth  of  (7")  at  the  point  P,  the  condition 
that  (7")  hyperosculate  c  at  P  is  that  the  equations, 

8.  (2/'  +  X)^2  +  (1  -  \y')li  =  0, 

9.  •       (1  +  y")h  +  2X(1  +  y")k  +  y%  -  X2/'7i  =  0, 

10.  -  2(1  +  y"){l  +  \y")k  +  2(1  +  y'){\  -  y')k 

+  (32/'2/"  -  \y")U  +  (62/'2/"X  +  2y")h  +  y"'l2  -  \y"'h  =  0, 

11.  -  I2y'y"{l  +  \y')k+  Wy"{\  -  y')h 

+  (32/"^  +  42/'2/'"  -  2X^"0^4 

+  (6X2/"'  +  SXy'y'"  +  4y"')h  +  r%  -  y^y'^^h  =  0, 
*  The  six  constants  ai,  aa,  /3i,  /Sj,  71,  72  are  now  connected  by  the  relation 

Ti^  +  72*  «i7i  +  "272 


14  Differential  Invariants  under 


12.  -i20yY'+d0y'"){l+\y') 
-  ^y"'{y'  -  X) 
-  ZQy"^ 


k+(20yy''+30y'"){\^y') 
+  4y"'a  +  \y') 
-  I2y"\ 
h  +  (52/'2/-  +  lOy'Y'  -  3X2/-)Z4 

+  (lOXi/'z/-  +  20X2/''2/"'  +  6^-)Z3  +  2/^/2  -  X^^i  =  0, 
shall  be  satisfied,  where 

^1  =  («i^i  +  a2^2)  -  71(^1'  +  ^2'), 
h  =  (aii82  -  «2)Si)  +  72()8i'  +  /322), 
h  =  7i(«ii82  —  Q!2iSi)  +  72(71^  +  72^^), 
^4  =  (ai^  +  a^')  -  W  +  132^)  (71'  +  72^), 

^5  =    (71^  +  72^)  (0:1182  -   OLi^l)  +  7l(«l^  +  Ot2^), 

k  =  (71'  +  72')(aii8i  +  a2^2)  -  7i(«i'  +  0C2'), 
and  where  the  new  constants  are  connected  by  the  relation 

(13)  hh  -  hk  +  hh  =  0. 

Solving  equations  8,  9,  10,  11,  12  for  the  ratios  of  the  Vs  and 
susbtituting  in  (13)  we  have 

(14)  ^_  4(1^25  =  0 

If  rp  is  the  angle  between  the  polar  axis  and  the  inflectional 

tangent  of  the  logarithmic  double  spiral  tan  ^  =  1/X  and  (14) 

becomes 

(14')  A5  +  8A33  Cot  2^  =  0. 

Equation  (14')  is  condition  under  which  a  logarithmic  equiangular 
double  spiral,  hyperosculate  curve  c  at  P,  or  stated  differently, 
equation  (14')  is  the  differential  equation  of  the  00^  logarithmic 
equiangular  double  spirals  (^  being  fixed)  expressed  symbolically 
in  terms  of  two  relative  differential  invariants  of  G^. 

Theorem  IV:  The  differential  equation  of  the  logarithmic  equi- 
angular double  spirals,  expressed  in  terms  of  the  invariants  of  G^,  is 
As  +  8A33  Cot  2iA  =  0. 


THE  Inveksion  Geoup.  15 

If  ^  =  45°,  A5  =  0  and  from  Theorem  IV  and  Corollary  I, 
Theorem  II,  we  have 

Corollary  I.  The  integral  curves  of  the  third  relative  differ- 
ential invariant  A5  =  0  are  the  curves  of  that  family  of  logarithmic 
equiangular  double  spirals  with  an  angle  of  45"  between  their  polar 
axis  and  their  inflectional  tangent;  or  the  integral  curves  of  A5  =  0 
are  the  inverse  transforms  of  those  logarithmic  spirals  cutting  their 
radii  vectors  at  an  angle  of  45°.* 

As  a  consequence  of  equation  (14')  and  equation  (6')  of 
section  1,  we  have 

Theorem  V.  At  any  point  P  on  a  logarithmic  double  spiral 
with  an  angle  xf/  between  the  polar  axis  and  the  inflectional  tangent, 
the  osculating  bicircular  quartic,  with  a  double  point  at  P,  has 
an  angle  6  between  the  branches,  such  that  d  =  Cot""^  f  Cot  2x1/. 

Corollary  I.  At  any  point  P  on  a  45°  logarithmic  double 
spiral,  the  branches  of  the  osculating  bicircular  quartic  having  a 
double  point  at  P  are  orthogonal. 

Corollary  II.  At  the  point  of  inflection  of  a  45°  logarithmic 
double  spiral  the  polar  axis  of  the  spiral  bisects  the  angle  between 
the  branches  of  the  osculating  bicircular  quartic  having  a  double 
point  at  the  point  of  osculation. 

III.  The  Differential  Parameter  for  G^  and  Differential 
Invariants  of  Order  Six. 

1.  In  order  to  determine  the  fundamental  differential  invari- 
ant of  the  sixth  order,  it  is  necessary  to  find  the  differential  param- 
eter for  Gq.  By  making  use  of  the  extension  of  the  independent 
infinitesimal  transformations,  as  far  as  the  fourth  order,  for  S 
of  part  one,  section  three,  and  then  extending  with  respect 
to  <p,  which  we  shall  assume  is  a  differential  invariant  under 
Ge,  we  have  the  complete  system  of  linear  partial  differential 

*  This  corollary  in  Inversion  Geometry  is  the  analog  of  one  of  Halphen's 
theorems  in  Projective  Geometry,  i.  e.,  "  Les  courbes  int^grales  de  I'equation 
A  =  0  sont  des  transform^es  homographiques  quelconques  de  la  spirale  log- 
arithmic qui  coupe  ses  rayons  sur  Tangle  de  30  degrees  "  A  is  the  third  relative 
differential  invariant  under  the  projective  group. 

Halphen: — Thesis:  Les  Invariants  Diff^rentiels. 


16  Differential  Invariants  under 

equations, 

df  df 

4-  Wy"'  +  ^y'")  -~r,  +  \hy'y^  +  \^"y"')  ^ 


+  ^vg.=  o. 


"^ax^^dy^^a/     ^  a/'     ^y   by'"     ^y  dy^- 


(,^-^)|-2.,|-2,(l  +  ,'^| 

T:        +  {2xy"  -  Qyy'y"  -  2y'{\  +  y'^)]  ^ 

+  [4a:2/'"  -  Qyy'"  -  Syy'y'"  -  WY']^r 

+  [Qxy-^-  -  lOyyY-  -  20yV  -  SOyY"  +  42/'"]  ^ 

-2{yy'-x)<p'^,=  0, 
2xyl+(f-^)f^-2xa  +  y'^)§ 

-  [2yy"  +  &xy'y"  +  2(1  +  y")]^, 

-  [W  +  ^xy"'  +  ^y'y'"  +  I2y'y"']-^, 

-  [62/2/-+  10a:^'2/-+  182/"2+20a;2/"2/'"+242/y"]  ^^ 


-2(2/  +  ar2/')^'^=0. 


THE  Inversion  Group.  17 

The  same  reduction  process  that  was  applied  to  S,  brings  T  to 
the  form 

(l  +  2/'')|i+(3r-22/y")^. 


2". 


+  iWY'  -  ^y'y'-)  ^.  -  y'<p'  ^  =  o, 
.      -2(i  +  2/-)|-,-i2,y'^ 

+  [42/' V"  -  202/y"  -  Uy'Y'  -  181/"^]^  =  0, 

and  T'  is  equivalent  to  the  involution  system. 

df      <p'[{2y'y"'  +  ^y"')iX  +  y")  -  l2y'Y']  df  _ 
dy'  2[1  +  y'V'il  +  y")  -  Zy'y"']       d<p'      "' 

^  •  %"    2/'"(i  +  2/")  -  32/y'=' a^'    "' 

a/     y'"(i  +  2/'=^)  -  32/y''  a/_Q^ 


=  c 


dy'"  <p'il  + 1/'=^)  d<p 

The  solution  of  T"  is  readily  obtained  and  is 

cp'd  +  y") 
y"(i  +  y")  -  SyY'Y" 

or  symbolically  expressed 

Since  the  differential  invariant  of  lowest  order,  As/As^  =  Is,. is 
known  (Part  I,  section  3),  we  put  tp  —  As/As^  and  by  making 
use  of  Ai{d<p/dx)/Az^'^  =  c,  the  fundamental  differential  invariant 
of  order  six  is  Ai[A3A5'  —  SAs'Asl/As^'^  =  h,  where  the  primes 
denote  differentiation  with  respect  to  the  dependent  variable  x. 
The  following  theorem  may  be  stated. 

Theorem  VI.     The  most  general  differential  invariant  of  order 


18  Differential  Invariants  under 

six  is  of  the  form 

-fA^    Ai[A6^A3-3A3^A5]'] 
•^LAs^'  A3«/2  J-c. 

In  order  to  give  a  geometric  interpretation  of  a  differential 
invariant  of  order  six,  we  may  make  use  of  a  bicircular  quartic 
of  deficiency  zero.    Let  M  be  the  discriminant  of 

Q:  {^  +  f)  +  {ax  +  by){x^  +  y')  +  cx^  +  2dxy  +  ef 

+  2fx  +  2gy+h  =  0. 
Since  Jf  is  a  function  of  the  coeflBcients,  a,h,  •  -  •  h,  we  may  write 
M  =  m{a,  b,  c,  d,  e,f,  g,  h). 

Let  m  =  0,  then  Q  is  of  deficiency  zero,  and  Q,  subject  to  the 
condition  m  =  0,  may  have  contact  of  order  six  with  c  (part 
two,  section  one)  at  P.  The  coeflBcients  a,h,  -  •  -  g  are  expressible 
in  terms  of  y',  y",  •  •  •  y^K  Since  it  is  known  that  angles  remain 
invariant  under  Ge,  we  have 

Theorem  VII.  7/  a  bicircular  quartic  of  deficiency  zero,  osculate 
a  curve  c  at  a  point  P  (c  being  analytic  in  the  neighborhood  of  P) 
the  angle  between  the  branches  of  the  real  node  of  Q  is  expressible  as 
a  function  of  1 5  and  Iq. 

2.  It  is  interesting  to  note  that  the  differential  equation  of 
the  00^  logarithmic  double  spirals,  arises  in  connection  with  a 
certain  family  of  bicircular  quartics,  and  that  this  class  of  double 
spirals  are  made  up  of  a  special  kind  of  points,  which  are  closely 
analogous  to  Halphen's  points  of  coincidence.  This  is  brought 
out  as  follows.  Let  c  be  a  curve  analytic  in  the  neighborhood  of 
point  P,  and  let  U  and  V  be  two  bicircular  quartics,  each  having 
contact  with  c  at  P  of  order  six.  The  family  of  bicircular 
quartics  have  fifteen  points  in  common,  i.  e.,  seven  points  at  P 
and  four  points  at  each  of  the  circular  points  at  infinity.  Besides 
these  fifteen  points  of  intersection,  the  members  of  this  family 
have  a  sixteenth  point  in  common,  which  we  shall  call  P'.  It 
may  happen  that  P'  will  coincide  with  P.  If  such  is  the  case 
we  shall  speak  of  P  as  a  "  bicircular  quartic  point  of  coincidence."* 

*  The  "  bicircular  quartic  point  of  coincidence  in  Inversion  Geometry  is 
analogous  to  Halphen's  point  of  coincidence  in  Projective  Geometry. 


THE  Inversion  Group.  19 

Among  the  penosculating*  bicircular  quartics  of  the  family 
U  +  W,  there  will  be  one  member  of  the  family,  let  us  say  W, 
having  a  double  point  at  P  and  one  of  the  branches  of  W  will 
have  contact  of  order  five  with  c  at  P  and  with  each  member  of 
the  family  U  -\-W.  If  then  W  should  have  contact  of  order  six 
with  another  one  of  the  family,  say  U,  the  sixteenth  point  of 
intersection  of  the  members  of  U  -{-  W,  will  coincide  with  P. 
Conversely,  if  the  sixteenth  point  of  intersection  of  U  -^W 
should  coincide  with  P,  then  one  branch  of  W  will  have  contact 
of  order  six  with  each  member  of  the  family  U  -{-W. 

Theorem  VIII.  The  necessary  and  sufficient  condition  jor  the 
existence  of  a  "  bicircular  quartic  point  of  coincidence  "  at  a  point 
P  on  a  curve  C  is  that  the  osculating  bicircular  quartic,  having  a 
double  point  at  P,  have  contact  with  c  of  order  six. 

To  express  the  content  of  the  above  theorem  analytically,  we 
assume  that  P  is  the  origin,  and  that  W,  having  a  double  point 
at  P,  has  contact  with  c  of  order  six.    The  equation  of  W  is 

W:   (x^  +  2/2)2  _|_  (^3.  +  52/)(a;2  +  y^)  +  cx^  +  2dxy  -\- ef  =  0 

subject  to  the  conditions 

15.  c  +  2y'd  +  y'^e  =  0, 

16.  (1  +  y")a  +  2/'(l  +  y'')b  +  y"d  +  yY'e  =  0, 

17.  12(1  +  y'y  +  I2y'y"a  +  [62/"  +  WY]b 

+  ^y"'d  +  W  +  ^y'y"']e  =  0, 

(18)  2Ay'y"iX  +  v")  +  Wv'"  +  ^y'> 

+  [2y"'  +  ^y'Y'  +  ^y'y"']h  +  y'^^d^WY'+y'r^le  =  0, 

19.  2402/'2/"'(l  +  y'^)  +  180y'"(l  +  ^") 

+  myY  +  60i/"t/'"]a  +  [152/-  +  ^^y'^ 
+  imy'y"y"'  +  ^by"^]b 

+  62/vcZ  +  [Iby'Y  +  IO2/""  +  62/'2/v]e  =  0, 

*  Prof.  E.  J.  Wilczynski  has  used  the  term  '  penosculants '  to  denote  a 
class  of  curves  for  which  the  order  of  contact  falls  short  of  the  maximum  by  a 
single  unit. 


20  Differential  Invariants  under 

20.  60y'2/-(i  +  y'^  +  noy'Y'a  +  ^y") 

+  \Wy"'  +  Wr  +  152/''^-  +  102/"'']a 

+  [3r  +  92/'y  +  ^hy'y"r^  +  452/"^'"  +  302/'2/""]6 

+  y'd  +  [32/'V  +  52/'"2/J^  +  2/y ']e  =  0, 

and  this  set  of  linear  equations  is  satisfied  if  and  only  if  the  de- 
terminant D  vanishes,  the  columns  of  D  being 

(1)  0 

0 

12(1  +  yy 

2^y'y"{\  +  y"^ 

240^y"(l  +  2/")  +  1802/"^(1  +  y") 

602/'2/-(l  +  y'^  +  l20y"y"'iX  +  32/")  +  ISOz/Y'' 

(2) 


(3) 


(4,  5,  6) 


0 

i  +  y" 

Wy" 

^y'y'"  +  32/"^ 

ZOyY-  +  602/''^"' 

62/y  +  iW'y'^  +  102/"" 

0 

2/'(l  +  y") 

62/"  +  182/'y' 

2y"'  +  62/'y "  +  92/'2/"' 

15^'^ 

+  452/'V'^  +  1802/'2/'y "  +  452/'" 

3r  +  92/'y  +  452/'^"^^-  +  452/"V"  +  2>0y'y"" 

\            1 

2^'                       2/", 

0 

2/"             2/y', 

0 

42/'"              32/"'  H-  42/'2/"', 

0 

2/iv             2y"y"'  +  2/'2/*", 

0 

62/-      152/"2/-  4-  IO2/""  +  62/'2/^ 

0 

2/vi     3y'y  _|.  5y'"2/'"  +  2/'2/"' 

Geometrically  it  is  evident  that  D  is  of  the  form  J{If>,  h)-    Upon 
evaluation  and  substitution  we  have 

D  =  SAiAslAaAg'  -  3A3'A5] 


THE  Inversion  Group.'-,  ;    ',  l^ ''';].' \'\\it\ 

and  the  necessary  and  sufficient  condition  for  a  "  bicircular 
quartic  point  of  coincidence/'  analytically  expressed,  is 

(21)  A3A5'  -  SAs'As*  =  0. 

Again  consider  the  bicircular  quartic 

Q:  {x'  +  y')  +  (ax  +  hy){x''  +  y')  +  cx^  +  2dxy 

+  ey^  +  2fx  -\-2gy+h  =  0. 

If  at  the  point  P,  (0,  0),  Q  be  given  contact  of  order  seven  with 
the  curve  c,  i,  e.,  if  Q  osculate  c  at  P,  the  coefficients  of  Q  can  be 
expressed  in  terms  of  y',  y",  •  •  •  y""".  On  expressing  the  con- 
dition for  osculation  and  solving  for  g,  and  h,  it  is  found  that 
^  =  gr  =  0,  if  AsAs'  —  3A3'A5  =  0,  and  we  have  as  a  corollary  to 
theorem  VIII. 

Corollary  I.  At  a  "  bicircular  quartic  point  of  coincidence  " 
the  osculating  bicircular  quartic  has  a  double  point  at  the  point  of 
osculation. 

If  rp  be  eliminated  from  equation  (14')  by  differentiation, 
equation  (21)  is  obtained  and  we  have 

Corollary  II.  The  logarithmic  double  spirals  are  curves,  all 
of  whose  points  are  "  bicircular  quartic  points  of  coincidence." 

Corollary  III.  At  any  point  on  a  logarithmic  double  spiral, 
the  osculating  bicircular  quartic  has  a  double  point  at  the  point  of 
osculation. 

In  the  last  two  corollaries  we  have  new  properties  of  the  logar- 
ithmic double  spiral. 

IV.    Differential    Invariants    Intrinsically    Expressed. 
Differential  Invariants  of  Order  Higher  than  the 

Sixth. 

1.  According  to  the  change  of  variables  in  part  one,  section 
three,  the  first  absolute  differential  invariant  may  be  expressed  as 

^^^^  {dp/dsy  ~^'' 

*  Ai  =  0  and  A3  =  0  appear  as  extraneous  solutions  in  finding  the  condition 
for  such  points. 


"22:  ■  Differential  Invariants 

where  p  is  the  radius  of  curvature  and  5  is  the  arc  length.  Simi- 
larly the  differential  parameter  for  G^  may  be  written 

p{d(p/ds)/{dp/ds). 

The  intrinsic  differential  equation  of  the  oo^  logarithmic  equi- 
angular double  spirals  is 

Equation  (23)  expressed  in  terms  of  the  first  four  successive  radii 
of  curvature  p,  pi,  p2  and  pi,  becomes 

(24)  4p3Pip2  -  10p2Pi2p  -  5p2V  +  15pi^  -  4piV 

+  8piV  Cot  2x1/  =  0, 
and  we  have 

Theorem  IX.  At  any  point  on  a  logarithmic  equiangular 
double  spiral,  the  first  four  successive  radii  of  curvature  satisfy  the 
relation  expressed  by  equation  (24). 

2.  The  general  expression  for  the  fundamental  differential 
invariant  of  order  n  is  readily  found.  If  we  put  A3A5'— SAa'As 
=  06,  we  have  for  the  fundamental  invariant  of  order  six 
AiQ^3~^^l^\  Again  if  67  represents  the  numerator  of  the  fraction 
found  by  taking  the  derivative  of  h  with  respect  to  x,  we  have 
for  the  fundamental  invariant  of  order  seven 

AiOzAs"*. 

It  is  easy  to  generalize  this  process.  If  then  0„  be  the  numerator 
of  the  fraction  formed  by  taking  the  derivative,  of  the  funda- 
mental differential  invariant  of  the  (n  —  l)st  order,  with  respect 

to  X,  we  have 

AiGnAa-^^^-^^^  =  j^^ 

and  we  have 

Theorem  X.  The  most  general  differential  invariant  of  the 
rUh  order,  under  G^,  is  of  the  form 

and  every  invariant  family  of  <x>^  curves  is  the  solution  of  a  differ- 
ential equation  of  the  nth  order,  expressible  in  the  above  form. 


VITA. 

George  Walker  Mullins.  Born  Fayetteville,  Ark.,  Feb.  11, 
1881.  Graduate  University  of  Arkansas,  A.B.,  1904;  Columbia 
University,  M.A.,  1913.  Professor  of  Mathematics,  Simmons 
College,  Abilene,  Texas,  1905-1912.  Student  University  of 
Chicago,  Summer  Sessions,  1907, 1909, 1910.  Graduate  Student 
and  Instructor  in  Extension  Teaching,  Columbia  University, 
1912-1913.  Instructor  in  Mathematics,  Barnard  College,  Colum- 
bia University,  1913-.  Member  of  the  American  Mathematical 
Society. 

Grateful  acknowledgment  is  made  to  Prof.  Edward  Kasner 
for  his  helpful  suggestions  and  encouragement  in  the  prepara- 
tion of  this  paper. 


14  DAY  USE 

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